imported>Etone |
imported>Macdonald-ross |
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| [[File:Under_construction.png|60px]] <font color=red size =5>Under construction.</font>
| | [[File:Cartesian coordinate system handedness.svg|thumb|The left-handed orientation is shown on the left, and the right-handed on the right.]] |
| | [[File:Manoderecha.svg|thumb|Prediction of direction of field (''B'') when the current ''I'' flows in the direction of the thumb]] |
| | [[File:Right-hand grip rule.svg|thumb|The right-hand rule for motion produced with screw threads]] |
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| =Lovász Local Lemma=
| | The '''right-hand rule''' is a convention in [[vector]] math. It helps you remember [[direction]] when vectors get [[vector math|cross multiplied]]. |
| Assume that <math>A_1,A_2,\ldots,A_n</math> are "bad" events. We are looking at the "rare event" that none of these bad events occurs, formally, the event <math>\bigwedge_{i=1}^n\overline{A_i}</math>. How can we guarantee this rare event occurs with positive probability? The ''Lovász Local Lemma'' provides an answer to this fundamental question by using the information about the dependencies between the bad events.
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| ==The dependency graph ==
| | :# Start by closing your right hand and stick out your pointer finger. |
| The notion of mutual independence between an event and a set of events is formally defined as follows.
| | :# Stick your [[thumb]] straight up like a your making the sign for a [[gun]]. |
| {{Theorem|Definition (mutual independence)|
| | :# If you point your "gun" straight ahead, stick out your middle finger so that it points left and all your fingers are at [[right angle|right angles]] to each other. |
| :An event <math>A</math> is said to be '''mutually independent''' of events <math>B_1,B_2,\ldots, B_k</math>, if for any disjoint <math>I^+,I^-\subseteq\{1,2,\ldots,k\}</math>, it holds that | |
| ::<math>\Pr\left[A \mid \left(\bigwedge_{i\in I^+}B_i\right) \wedge \left(\bigwedge_{i\in I^-}\overline{B_i}\right)\right]=\Pr[A]</math>.
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| }}
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| Given a sequence of events <math>A_1,A_2,\ldots,A_n</math>, we use the '''dependency graph''' to describe the dependencies between these events.
| | If you have two vectors that you want to cross multiply, you can figure out the direction of the vector that comes out by pointing your thumb in the direction of the first vector and your pointer in the direction of the second vector. Your middle finger will point the direction of the cross product. |
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| {{Theorem
| | Remember that when you change the order that vectors get cross multiplied, the result goes in the opposite direction. So it's important to make sure that you go in the order of <math>\vec{thumb} \times \vec{pointer} = \vec{middle}</math>. |
| |Definition (dependency graph)|
| |
| :Let <math>A_1,A_2,\ldots,A_n</math> be a sequence of events. A graph <math>D=(V,E)</math> on the set of vertices <math>V=\{1,2,\ldots,n\}</math> is called a '''dependency graph''' for the events <math>A_1,\ldots,A_n</math> if for each <math>i</math>, <math>1\le i\le n</math>, the event <math>A_i</math> is mutually independent of all the events <math>\{A_j\mid (i,j)\not\in E\}</math>.
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| :Furthermore, for each event <math>A_i</math>:
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| :* define <math>\Gamma(A_i)=\{A_j\mid (i,j)\in E\}</math> as the '''neighborhood''' of event <math>A_i</math> in the dependency graph;
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| :* define <math>\Gamma^+(A_i)=\Gamma(A_i)\cup\{A_i\}</math> as the '''inclusive neighborhood''' of <math>A_i</math>, i.e. the set of events adjacent to <math>A_i</math> in the dependency graph, including <math>A_i</math> itself.
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| }}
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| ;Example
| | ==Variations== |
| :Let <math>X_1,X_2,\ldots,X_m</math> be a set of ''mutually independent'' random variables. Each event <math>A_i</math> is a predicate defined on a number of variables among <math>X_1,X_2,\ldots,X_m</math>. Let <math>v(A_i)</math> be the unique smallest set of variables which determine <math>A_i</math>. The dependency graph <math>D=(V,E)</math> is defined by
| | There is another rule called the right-hand grip rule (or corkscrew rule) that is used for [[magnetism|magnetic fields]] and things that [[rotation|rotate]]. |
| :::<math>(i,j)\in E</math> iff <math>v(A_i)\cap v(A_j)\neq \emptyset</math>. | | :# Start by putting your right hand out flat and point your thumb straight out so that it is at a [[right angle]] to your other [[fingers]]. |
| | :# Now curl your fingers into a [[fist]] and keep your [[thumb]] out (like a Thumbs Up). |
| | :# Match how your fingers curl in to the way something is moving. The direction that your thumb is pointing is the direction of the vector we use to talk about it. |
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| == The local lemma ==
| | You can do this in reverse by starting your thumb in the direction of the vector and see how your fingers curl to see the direction of rotation. If you point your thumb in the direction of current in a wire, the magnetic field that comes up around it is in the direction of your curling fingers. |
| The following lemma, known as the Lovász local lemma, first proved by Erdős and Lovász in 1975, is an extremely powerful tool, as it supplies a way for dealing with rare events.
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| {{Theorem
| | [[Category:Mathematics]] |
| |Lovász Local Lemma (symmetric case)|
| |
| :Let <math>A_1,A_2,\ldots,A_n</math> be a set of events, and assume that there is a <math>p\in[0,1)</math> such that the followings are satisfied:
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| :#for all <math>1\le i\le n</math>, <math>\Pr[A_i]\le p</math>; | |
| :#the maximum degree of the dependency graph for the events <math>A_1,A_2,\ldots,A_n</math> is <math>d</math>, and
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| :::<math>\mathrm{e}p\cdot (d+1)\le 1</math>.
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| :Then
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| ::<math>\Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right]>0</math>.
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| }}
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| The following is a general asymmetric version of the local lemma. This generalization is due to Spencer.
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| {{Theorem
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| |Lovász Local Lemma (general case)|
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| :Let <math>A_1,A_2,\ldots,A_n</math> be a sequence of events. Suppose there exist real numbers <math>x_1,x_2,\ldots, x_n</math> such that <math>0\le x_i<1</math> and for all <math>1\le i\le n</math>,
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| ::<math>\Pr[A_i]\le x_i\prod_{A_j\in \Gamma(A_i)}(1-x_j)</math>.
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| :Then
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| ::<math>\Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right]\ge\prod_{i=1}^n(1-x_i)</math>.
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| }}
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| To see that the general LLL implies symmetric LLL, we set <math>x_i=\frac{1}{d+1}</math> for all <math>i=1,2,\ldots,n</math>. Then we have <math>\left(1-\frac{1}{d+1}\right)^d>\frac{1}{\mathrm{e}}</math>.
| | {{math-stub}} |
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| Assume the condition in the symmetric LLL:
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| :#for all <math>1\le i\le n</math>, <math>\Pr[A_i]\le p</math>;
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| :#<math>\mathrm{e}p\cdot(d+1)\le 1</math>;
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| then it is easy to verify that for all <math>1\le i\le n</math>,
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| :<math>\Pr[A_i]\le p\le\frac{1}{\mathrm{e}(d+1)}<\frac{1}{d+1}\left(1-\frac{1}{d+1}\right)^d\le x_i\prod_{A_j\in\Gamma(A_i)}(1-x_j)</math>.
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| Due to the general LLL, we have
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| :<math>\Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right]\ge\prod_{i=1}^n(1-x_i)=\left(1-\frac{1}{d+1}\right)^n>0</math>.
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| This proves the symmetric LLL.
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| Alternatively, by setting <math>x_i=\frac{1}{d}</math> and assuming without loss of generality that the maximum degree of the dependency graph has <math>d\ge 2</math>, we can have another symmetric version of the local lemma.
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| {{Theorem
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| |Lovász Local Lemma (symmetric case, alternative form)|
| |
| :Let <math>A_1,A_2,\ldots,A_n</math> be a set of events, and assume that there is a <math>p\in[0,1)</math> such that the followings are satisfied:
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| :#for all <math>1\le i\le n</math>, <math>\Pr[A_i]\le p</math>;
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| :#the maximum degree of the dependency graph for the events <math>A_1,A_2,\ldots,A_n</math> is <math>d</math>, and
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| :::<math>4p d\le 1</math>.
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| :Then
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| ::<math>\Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right]>0</math>.
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| }}
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| The original proof of the Lovász Local Lemma is by induction. See [[高级算法 (Fall 2016)/Nonconstructive Proof of Lovász Local Lemma| this note]] for the original non-constructive proof of Lovász Local Lemma.
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| | |
| =Random Search for <math>k</math>-SAT=
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| We start by giving the definition of <math>k</math>-CNF and <math>k</math>-SAT.
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| {{Theorem|Definition (exact-<math>k</math>-CNF)|
| |
| :A logic expression <math>\phi</math> defined on <math>n</math> Boolean variables <math>x_1,x_2,\ldots,x_n\in\{\mathrm{true},\mathrm{false}\}</math> is said to be a '''conjunctive normal form (CNF)''' if:
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| :* <math>\phi</math> can be written as a conjunction(AND) of '''clauses''' as <math>\phi=C_1\wedge C_2\wedge\cdots\wedge C_m</math>;
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| :* each clause <math>C_i=\ell_{i_1}\vee \ell_{i_2}\vee\cdots\vee\ell_{i_k}</math> is a disjunction(OR) of '''literals''';
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| :* each literal <math>\ell_j</math> is either a variable <math>x_i</math> or the negation <math>\neg x_i</math> of a variable.
| |
| :We call a CNF formula '''<math>k</math>-CNF''', or more precisely an '''exact-<math>k</math>-CNF''', if every clause consists of ''exact'' <math>k</math> ''distinct'' literals.
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| }}
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| For example:
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| :<math>
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| (x_1\vee \neg x_2 \vee \neg x_3)\wedge (\neg x_1\vee \neg x_3\vee x_4)\wedge (x_1\vee x_2\vee x_4)\wedge (x_2\vee x_3\vee \neg x_4)
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| </math>
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| is a <math>3</math>-CNF formula by above definition.
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| | |
| ;Remark
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| :The notion of <math>k</math>-CNF defined here is slightly more restrictive than the standard definition of <math>k</math>-CNF, where each clause consists of ''at most'' <math>k</math> variables. See [https://en.wikipedia.org/wiki/Boolean_satisfiability_problem#3-satisfiability here] for a discussion of the subtle differences between these two definitions.
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| A logic expression <math>\phi</math> is said to be '''satisfiable''' if there is an assignment of values of true or false to the variables <math>\boldsymbol{x}=(x_1,x_2,\ldots,x_n)</math> so that <math>\phi(\boldsymbol{x})</math> is true. For a CNF <math>\phi</math>, this mean that there is an assignment that satisfies all clauses in <math>\phi</math> simultaneously.
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| The '''<math>k</math>-satisfiability (<math>k</math>-SAT)''' problem is that given as input a <math>k</math>-CNF formula <math>\phi</math> decide whether <math>\phi</math> is satisfiable.
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| {{Theorem|<math>k</math>-SAT|
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| :'''Input:''' a <math>k</math>-CNF formula <math>\phi</math>.
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| :Determines whether <math>\phi</math> is satisfiable.
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| }}
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| It is well known that <math>k</math>-SAT is '''NP-complete''' for any <math>k\ge 3</math>.
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| == Satisfiability of <math>k</math>-CNF==
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| As in the Lovasz local lemma, we consider the ''dependencies'' between clauses in a CNF formula.
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| We say that a CNF formula <math>\phi</math> has '''maximum degree''' at most <math>d</math> if every clause in <math>\phi</math> shares variables with at most <math>d</math> other clauses in <math>\phi</math>.
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| By the Lovasz local lemma, we almost immediately have the following theorem for the satisfiability of <math>k</math>-CNF with bounded degree.
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| {{Theorem|Theorem|
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| :Let <math>\phi</math> be a <math>k</math>-CNF formula with maximum degree at most <math>d</math>. If <math>d\le 2^{k-2}</math> then <math>\phi</math> is always satisfiable.
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| }}
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| {{Proof|
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| Let <math>X_1,X_2,\ldots,X_n</math> be Boolean random variables sampled uniformly and independently from <math>\{\text{true},\text{false}\}</math>. We are going to show that <math>\phi</math> is satisfied by this random assignment with positive probability. Due to the probabilistic method, this will prove the existence of a satisfying assignment for <math>\phi</math>.
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| Suppose there are <math>m</math> clauses <math>C_1,C_2,\ldots,C_m</math> in <math>\phi</math>. Let <math>A_i</math> denote the bad event that <math>C_i</math> is not satisfied by the random assignment <math>X_1,X_2,\ldots,X_n</math>. Clearly, each <math>A_i</math> is dependent with at most <math>d</math> other <math>A_j</math>'s, which means the maximum degree of the dependency graph for <math>A_1, A_2, \ldots, A_m</math> is at most <math>d</math>.
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| Recall that in a <math>k</math>-CNF <math>\phi</math>, every clause <math>C_i</math> consists of precisely <math>k</math> variable, and <math>C_i</math> is violated by only one assignment among all <math>2^k</math> assignments of the <math>k</math> variables in <math>C_i</math>. Therefore, the probability of <math>C_i</math> being violated is <math>p=\Pr[A_i]=2^{-k}</math>.
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| If <math>d\le 2^{k-2}</math>, that is, <math>4pd\le 1</math>, then due to Lovasz local lemma (symmetric case, alternative form), it holds that
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| :<math>\Pr\left[\bigwedge_{i=1}^m\overline{A_i}\right]>0</math>.
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| The existence of satisfying assignment follows by the probabilistic method.
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| }}
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| == Moser's recursive fix algorithm ==
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| The above theorem basically says that for a CNF if every individual clause is easy to satisfy and is dependent with few other clauses then the CNF should be always satisfiable. However, the theorem only states the existence of a satisfying solution, but does not gives a way to find such solution.
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| In 2009, Moser gave a very simple randomized algorithm which efficiently finds a satisfying assignment with high probability under the condition <math>d\le 2^{k-5}</math>.
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| We need the following notations. Given as input a CNF formula <math>\phi</math>:
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| * let <math>\mathcal{X}=\{x_1,x_2,\ldots,x_n\}</math> be the Boolean variables on which <math>\phi</math> is defined, and <math>\mathcal{C}</math> the set of clauses in <math>\phi</math>;
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| * for each clause <math>C\in \mathcal{C}</math>, we denote by <math>\mathsf{vbl}(C)\subseteq\mathcal{X}</math> the set of variables on which <math>C</math> is defined;
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| * we also abuse the notation and denote by <math>\Gamma(C)=\{D\in\mathcal{C}\mid D\neq C, \mathsf{vbl}(C)\cap\mathsf{vol}(D)\neq\phi\}</math> the '''neighborhood''' of <math>C</math>, i.e. the set of ''other'' clauses in <math>\phi</math> that shares variables with <math>C</math>, and <math>\Gamma^+(C)=\Gamma(C)\cup\{C\}</math> the '''inclusive neighborhood''' of <math>C</math>, i.e. the set of all clauses, including <math>C</math> itself, that share variables with <math>C</math>.
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| The algorithm consists of two components: the main function ''Solve''() and a sub-routine ''Fix''().
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| {{Theorem
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| |Solve(CNF <math>\phi</math>)|
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| :Pick values of <math>x_1,x_2\ldots,x_n</math> uniformly and independently at random;
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| :While there is an unsatisfied clause <math>C</math> in <math>\phi</math>
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| :: '''Fix'''(<math>C</math>);
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| }}
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| The sub-routine '''Fix()''' is a recursive procedure:
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| {{Theorem
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| |Fix(Clause <math>C</math>)|
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| :Replace the values of variables in <math>\mathsf{vbl}(C)</math> with new uniform and independent random values;
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| :While there is ''unsatisfied'' clause <math>D\in\Gamma^+(C)</math>
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| :: '''Fix'''(<math>D</math>);
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| }}
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| It is an amazing discovery that this simple algorithm works well as long as the condition of Lovasz local lemma is satisfied. Here we prove a slightly weaker statement for the convenience of analysis.
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| {{Theorem|Theorem|
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| :Let <math>\phi</math> be a <math>k</math>-CNF formula with maximum degree at most <math>d</math>.
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| :There is a universal constant <math>c>0</math>, such that if <math>d< 2^{k-c}</math> then the algorithm ''Solve''(<math>\phi</math>) finds a satisfying assignment for <math>\phi</math> in time <math>O(n+km\log m)</math> with high probability.
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| }}
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| The analysis is based on a technique called '''''entropy compression'''''. This is a very clever idea and may be very different from what you might have seen so far about algorithm analysis.
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| We first give a high-level description of this idea:
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| * We use <math>\mathsf{Alg}(r, \phi)</math> to abstractly denote an algorithm <math>\mathsf{Alg}</math> running on an input <math>\phi</math> with random bits <math>r\in\{0,1\}^*</math>. For an algorithm with no access to the random bits <math>r</math>, once the input <math>\phi</math> is fixed, the behavior of the algorithm as well as its output is ''deterministic''. But for ''randomized'' algorithms, the behavior of <math>\mathsf{Alg}(r, \phi)</math> is a random variable even when the input <math>\phi</math> is fixed.
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| * Fix an arbitrary (worst-case) input <math>\phi</math>. We try to construct a ''succinct representation'' <math>c</math> of the behavior of <math>\mathsf{Alg}(r, \phi)</math> in such a manner that the random bits <math>r</math> can always be fully recovered from this succinct representation <math>c</math>. In other words, <math>\mathsf{Alg}(r, \phi)</math> gives an encoding (a 1-1 mapping) of the random bits <math>r</math> to a succinct representation <math>c</math>.
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| * It is a fundamental law that random bits cannot be compressed significantly by any encoding. Therefore if a longer running time of <math>\mathsf{Alg}(r, \phi)</math> would imply that the random bits <math>r</math> can be encoded to a succinct representation <math>c</math> which is much shorter than <math>r</math>, then we prove the running time of the algorithm <math>\mathsf{Alg}(r, \phi)</math> cannot be too long.
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| :* A natural way to reach this last contradiction is to have the following situation: As the running time of <math>\mathsf{Alg}(r, \phi)</math> grows, naturally both lengths of random bits <math>r</math> and the succinct representation <math>c</math> of the behavior of <math>\mathsf{Alg}(r, \phi)</math> grow. So if the former grows much faster than the latter as the running time grows, then a large running time may cause the length of <math>r</math> significantly greater than the length of <math>c</math>.
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| ---------
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| We now proceed to the analysis of <math>\text{Solve}(\phi)</math> and prove the above theorem.
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| From now on we assume that the input instance <math>\phi</math> is an arbitrarily fixed exact-<math>k</math>-CNF formula with maximum degree <math>d</math>:
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| *Let <math>T</math> denote the total number of time the function <math>\text{Fix}()</math> is being called (including both the calls in <math>\text{Solve}(\phi)</math> and the recursive calls).
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| *Let <math>t=\min\{T,2^m\}</math>.
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| Note that <math>T</math> and <math>t</math> are random variables which depend solely on the random bits used by the algorithm (after the input <math>\phi</math> being fixed).
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| We are going to show that <math>t=O(m\log m)</math> with high probability, which means <math>T=O(m\log m)</math> with high probability.
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| The reason we consider <math>t=\min\{T,2^m\}</math> is that we are not sure whether <math>T</math> is finite or infinite, so we "truncate" <math>T</math> if it becomes too large. This truncation threshold is quite arbitrary as long as it is significantly greater than <math>m\log m</math>.
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| We start by making the following simple observations:
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| *A clause <math>C</math> in <math>\phi</math> will be satisfied after '''Fix'''(<math>C</math>) returned and will remain as being satisfied afterwards.
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| *At the moment a '''Fix'''(<math>C</math>) being called, the clause <math>C</math> must be unsatisfied.
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| {{Theorem|Proposition|
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| # There are at most <math>m</math> top-level callings to '''Fix'''(), where <math>m</math> is the total number of clauses in <math>\phi</math>.
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| # If a '''Fix'''(<math>C</math>) is being called, the values of variables in <math>C</math> are uniquely determined.
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| }}
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| ----------
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| We consider a restrictive case.
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| Let <math>X_1,X_2,\ldots,X_m\in\{\mathrm{true},\mathrm{false}\}</math> be a set of ''mutually independent'' random variables which assume boolean values. Each event <math>A_i</math> is an AND of at most <math>k</math> literals (<math>X_i</math> or <math>\neg X_i</math>). Let <math>v(A_i)</math> be the set of the <math>k</math> variables that <math>A_i</math> depends on. The probability that none of the bad events occurs is
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| :<math>
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| \Pr\left[\bigwedge_{i=1}^n \overline{A_i}\right].
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| </math>
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| In this particular model, the dependency graph <math>D=(V,E)</math> is defined as that <math>(i,j)\in E</math> iff <math>v(A_i)\cap v(A_j)\neq \emptyset</math>.
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| Observe that <math>\overline{A_i}</math> is a clause (OR of literals). Thus, <math>\bigwedge_{i=1}^n \overline{A_i}</math> is a '''<math>k</math>-CNF''', the CNF that each clause depends on <math>k</math> variables.
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| The probability
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| :<math>
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| \bigwedge_{i=1}^n \overline{A_i}>0
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| </math>
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| means that the the <math>k</math>-CNF <math>\bigwedge_{i=1}^n \overline{A_i}</math> is satisfiable.
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| The satisfiability of <math>k</math>-CNF is a hard problem. In particular, 3SAT (the satisfiability of 3-CNF) is the first '''NP-complete''' problem (the Cook-Levin theorem). Given the current suspect on '''NP''' vs '''P''', we do not expect to solve this problem generally.
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| However, the condition of the Lovasz local lemma has an extra assumption on the degree of dependency graph. In our model, this means that each clause shares variables with at most <math>d</math> other clauses. We call a <math>k</math>-CNF with this property a <math>k</math>-CNF with bounded degree <math>d</math>.
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| Therefore, proving the Lovasz local lemma on the restricted forms of events as described above, can be reduced to the following problem:
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| ;Problem
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| :Find a condition on <math>k</math> and <math>d</math>, such that any <math>k</math>-CNF with bounded degree <math>d</math> is satisfiable.
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| In 2009, Moser comes up with the following procedure solving the problem. He later generalizes the procedure to general forms of events. This not only gives a beautiful constructive proof to the Lovasz local lemma, but also provides an efficient randomized algorithm for finding a satisfiable assignment for a number of events with bounded dependencies.
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| Let <math>\phi</math> be a <math>k</math>-CNF of <math>n</math> clauses with bounded degree <math>d</math>, defined on variables <math>X_1,\ldots,X_m</math>. The following procedure find a satisfiable assignment for <math>\phi</math>.
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| {{Theorem
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| |Solve(<math>\phi</math>)|
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| :Pick a random assignment of <math>X_1,\ldots,X_m</math>.
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| :While there is an unsatisfied clause <math>C</math> in <math>\phi</math>
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| :: '''Fix'''(<math>C</math>).
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| }}
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| The sub-routine '''Fix''' is defined as follows:
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| {{Theorem
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| |Fix(<math>C</math>)|
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| :Replace the variables in <math>v(C)</math> with new random values.
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| :While there is unsatisfied clause <math>D</math> that <math>v(C)\cap v(D)\neq \emptyset</math>
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| :: '''Fix'''(<math>D</math>).
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| }}
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| The procedure looks very simple. It just recursively fixes the unsatisfied clauses by randomly replacing the assignment to the variables.
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| We then prove it works.
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| ===Number of top-level callings of Fix ===
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| In '''Solve'''(<math>\phi</math>), the subroutine '''Fix'''(<math>C</math>) is called. We now upper bound the number of times it is called (not including the recursive calls).
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| Assume '''Fix'''(<math>C</math>) always terminates.
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| :;Observation
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| ::Every clause that was satisfied before '''Fix'''(<math>C</math>) was called will still remain satisfied and <math>C</math> will also be satisfied after '''Fix'''(<math>C</math>) returns.
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| The observation can be proved by induction on the structure of recursion. Since there are <math>n</math> clauses, '''Solve'''(<math>\phi</math>) makes at most <math>n</math> calls to '''Fix'''.
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| We then prove that '''Fix'''(<math>C</math>) terminates.
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| === Termination of Fix ===
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| The idea of the proof is to '''reconstruct''' a random string.
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| Suppose that during the running of '''Solve'''(<math>\phi</math>), the '''Fix''' subroutine is called for <math>t</math> times (including all the recursive calls).
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| Let <math>s</math> be the sequence of the random bits used by '''Solve'''(<math>\phi</math>). It is easy to see that the length of <math>s</math> is <math>|s|=m+tk</math>, because the initial random assignment of <math>m</math> variables takes <math>m</math> bits, and each time of calling '''Fix''' takes <math>k</math> bits.
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| We then reconstruct <math>s</math> in an alternative way.
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| Recall that '''Solve'''(<math>\phi</math>) calls '''Fix'''(<math>C</math>) at top-level for at most <math>n</math> times. Each calling of '''Fix'''(<math>C</math>) defines a recursion tree, rooted at clause <math>C</math>, and each node corresponds to a clause (not necessarily distinct, since a clause might be fixed for several times). Therefore, the entire running history of '''Solve'''(<math>\phi</math>) can be described by at most <math>n</math> recursion trees.
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| :;Observation 1
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| ::Fix a <math>\phi</math>. The <math>n</math> recursion trees which capture the total running history of '''Solve'''(<math>\phi</math>) can be encoded in <math>n\log n+t(\log d+O(1))</math> bits.
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| Each root node corresponds to a clause. There are <math>n</math> clauses in <math>\phi</math>. The <math>n</math> root nodes can be represented in <math>n\log n</math> bits.
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| The smart part is how to encode the branches of the tree. Note that '''Fix'''(<math>C</math>) will call '''Fix'''(<math>D</math>) only for the <math>D</math> that shares variables with <math>C</math>. For a k-CNF with bounded degree <math>d</math>, each clause <math>C</math> can share variables with at most <math>d</math> many other clauses. Thus, each branch in the recursion tree can be represented in <math>\log d</math> bits. There are extra <math>O(1)</math> bits needed to denote whether the recursion ends. So totally <math>n\log n+t(\log d+O(1))</math> bits are sufficient to encode all <math>n</math> recursion trees.
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| :;Observation 2
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| ::The random sequence <math>s</math> can be encoded in <math>m+n\log n+t(\log d+O(1))</math> bits.
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| With <math>n\log n+t(\log d+O(1))</math> bits, the structure of all the recursion trees can be encoded. With extra <math>m</math> bits, the final assignment of the <math>m</math>
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| variables is stored.
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| We then observe that with these information, the sequence of the random bits <math>s</math> can be reconstructed from backwards from the final assignment.
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| The key step is that a clause <math>C</math> is only fixed when it is unsatisfied (obvious), and an unsatisfied clause <math>C</math> must have exact one assignment (a clause is OR of literals, thus has exact one unsatisfied assignment). Thus, each node in the recursion tree tells the <math>k</math> random bits in the random sequence <math>s</math> used in the call of the Fix corresponding to the node. Therefore, <math>s</math> can be reconstructed from the final assignment plus at most <math>n</math> recursion trees, which can be encoded in at most <math>m+n\log n+t(\log d+O(1))</math> bits.
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| The following theorem lies in the heart of the '''Kolmogorov complexity'''. The theorem states that random sequence is '''incompressible'''.
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| {{Theorem
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| |Theorem (Kolmogorov)|
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| :For any encoding scheme , with high probability, a random sequence <math>s</math> is encoded in at least <math>|s|</math> bits.
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| }}
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| Applying the theorem, we have that with high probability,
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| :<math>m+n\log n+t(\log d+O(1))\ge |s|=m+tk</math>.
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| Therefore,
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| :<math>
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| t(k-O(1)-\log d)\le n\log n.
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| </math>
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| In order to bound <math>t</math>, we need
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| :<math>k-O(1)-\log d>0</math>,
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| which hold for <math>d< 2^{k-\alpha}</math> for some constant <math>\alpha>0</math>. In fact, in this case, <math>t=O(n\log n)</math>, the running time of the procedure is bounded by a polynomial!
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| === Back to the local lemma ===
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| We showed that for <math>d<2^{k-O(1)}</math>, any <math>k</math>-CNF with bounded degree <math>d</math> is satisfiable, and the satisfied assignment can be found within polynomial time with high probability. Now we interprete this in a language of the local lemma.
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| Recall that the symmetric version of the local lemma:
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| {{Theorem
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| |Theorem (The local lemma: symmetric case)|
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| :Let <math>A_1,A_2,\ldots,A_n</math> be a set of events, and assume that the following hold:
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| :#for all <math>1\le i\le n</math>, <math>\Pr[A_i]\le p</math>;
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| :#the maximum degree of the dependency graph for the events <math>A_1,A_2,\ldots,A_n</math> is <math>d</math>, and
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| :::<math>ep(d+1)\le 1</math>.
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| :Then
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| ::<math>\Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right]>0</math>.
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| }}
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| Suppose the underlying probability space is a number of mutually independent uniform random boolean variables, and the evens <math>\overline{A_i}</math> are clauses defined on <math>k</math> variables. Then,
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| :<math>
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| p=2^{-k}
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| </math>
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| thus, the condition <math>ep(d+1)\le 1</math> means that
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| :<math>
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| d<2^{k}/e
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| </math>
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| which means the Moser's procedure is asymptotically optimal on the degree of dependency.
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